Symbolic algorithm of the functional-discrete method for a Sturm-Liouville problem with a polynomial potential. (English) Zbl 1412.65059
Summary: A new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete method is developed and justified for the Sturm-Liouville problem on a finite interval for the Schrödinger equation with a polynomial potential and the boundary conditions of Dirichlet type. The algorithm of the general scheme of our method is developed when the potential function is approximated by the piecewise-constant function. Our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis. The number of summands in these decompositions depends on the degree of the potential polynomial and on the correction number. Our method uses the algebraic operations only and does not need solutions of any boundary value problems and computations of any integrals unlike the previous version. A numerical example illustrates the theoretical results.
MSC:
| 65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 34B24 | Sturm-Liouville theory |
| 34L16 | Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators |
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
Keywords:
eigenvalue problem; Sturm-Liouville problem; polynomial potential; functional-discrete method; symbolic algorithm; super-exponentially convergence rateReferences:
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